There was a time when I was absolutely driven to understand plane symmetry especially as regards knitted fabric. I created a database in FileMaker, including dozens of illustrations I made (pixel by pixel) and from it ran a report that created this pdf file. I was trying to enumerate all the different kinds of symmetry that are possible with knitting motifs using two colors.
I only used symmetries that can be fairly faithfully shown on a fabric that is based on stitches that lie on a rectangular grid.It probably helps, to understand this a little, what the wallpaper symmetry groups are. But it's pretty even without knowing them.
Two books I used to derive this were Symmetries of Culture and Tilings and Patterns.
